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Rapid thermalization and thermal radiation: From CGC to QGP. Kirill Tuchin in collaboration with Dima Kharzeev and Genya Levin. 2006 RHIC & AGS Annual Users’ Meeting. The main puzzle at RHIC. Hydrodynamic models are successful if thermalization time is Δt≈0.5 fm Naively, Δt∼1/(nσ)∼fm/α s.
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Rapid thermalization and thermal radiation: From CGC to QGP • Kirill Tuchin • in collaboration with Dima Kharzeev and Genya Levin 2006 RHIC & AGS Annual Users’ Meeting
The main puzzle at RHIC • Hydrodynamic models are successful if thermalization time is Δt≈0.5 fm • Naively, Δt∼1/(nσ)∼fm/αs (Baier, Mueller, Son, Schiff) Why thermalization is so fast?
Possible solutions • Plasma instabilities (Mrowczynski; Arnold, Lenaghan, Moore). • Early isotropization may help explain v2 (Arnold, Lenaghan, Moore; Rebhan, Romatschke, Strickland). • No thermalization in pQCD (Kovchegov). • AdS/CFT ☺
Space-time structure of the HIC t x- x+ • Upon collision heavy ions experience very large deceleration. • Indeed, most of matter is produced in the transverse plane! Aμ z • How large is acceleration? • It depends only on the strength of the nuclear field ⇒ Limiting fragmentation.
Comoving frame t x- x+ • Comoving frame: Aμ z • ρ, η define the Rindler space in which accelerated particle is at rest. • In the Rindler space x- and x+ are event horizons. Therefore, the produced particle can carry information only about conserved quantities. This is realized in the thermal distribution with T=a/(2π) (Unruh,Hawking).
Hagedorn argument • Consider breakdown of a high energy hadron of mass m into a final hadronic state of mass M. Transition probability: • In dual resonance model density of states is • Transition amplitude: • Probability conservation requires that ∑M P(M) = 1.
Limiting temperature • Therefore the string tension cannot accelerate particles beyond which exactly corresponds to the Hagedorn temperature: • One needs stronger fields to get T>TH; e.g. CGC • In this macroscopic quasiclassical picture one can also study phase transitions. (Ohsaku)
Particle production at high energies • Structure of the partonic cascade at high energies: A rapidity p,A,γ* time τ
Background field method • Structure of the partonic cascade at high energies: A static field sources “CGC” rapidity p time • Strength of the field is determined by the density of color charges in the transverse plane, i.e. Qs(y)
Field configuration at low x (I) Kharzeev, KT hep-ph/051234; Kharzeev, Levin, KT hep-ph/0602063 • (Chromo) Electric and Magnetic fields of a high energy hadron/nucleus are transverse (plane wave) and stable. • However, this is true only for a non-interacting hadron. Boundary conditions at the interaction point generate the longitudinal fields. • Recall reflection of light in Electrodynamics!
Field configuration at low x (II) • Since Qs(y) increases down the cascade, the parton transverse size decreases. Therefore, • and A(x,t) does not depend on s • Fries, Kapusta and Li argue that non-Abelian interactions can produce both longitudinal chromoelectric and chromomagnetic fields. • implications: see paper by Lappi & McLerran
Supercritical fields • The work done by the external chromo-electric field E accelerating a virtual qq pair apart by a Compton wavelength λc=h/mc is W=gEh/mc. • If W>2mc2 the pair becomes real. • In QED Ecr=1016 V/cm - beyond the current lab frontier. • In QCD g∼1, m∼Qs or Λ, thus Ecr∼(1 GeV)2 : pair production is a common phenomenon.
Vacuum rearrangement (I) • Example: charged scalar field in time dependent electromagnetic background • Eigenstates: where • Note, that ϕin = ϕ(t→-∞) are different from ϕout =ϕ(t→+∞)
Vacuum rearrangement (II) • Second quantization where • Equivalently • Unitarity implies • Therefore, even if
Particle production by black hole • The same problem of quantization in external field • In Schwartzschild metric • Black hole emission rate Hawking, 75 where the surface gravity is
Propagator in external field • Pair production rate = imaginary part of the propagator • WKB approximation is quite useful for calculating tunneling probabilities. • see T. Lappi’s numerical solution of Dirac equation in external field.
WKB approximation • In the pair production process electron’s energy changes from ε- to ε+ where • Pair production rate Γ≈e-2ImS. Imaginary part of action ImS can be found by integrating phase over states with imaginary momentum: where the turning points of the linear potential are
Canonical approach • Gauge boson propagator in SU(2) can be found from equation of motion • In the quasiclassical approximation we seek solution in form W=exp(-iS-ip⋅x) such that S’’<<(S’)2 • Introduce canonical momenta Pμ=-∂μS+gAμ. Then S can be found along the classical particle trajectories.
Action as a function of coordinates • Trajectory of a particle moving with constant acceleration coincides with the that of particle in a constant field E • Quasiclassical trajectory • Action • Introduce new variables • Imaginary part of action
Schwinger vs Unruh • ImS corresponds to instable, classically forbidden motion. Vacuum decay probability is Γ=exp(-2ImS). • In the constant field • How to reconcile this with Unruh thermal emission? • In Schwiner formula we average over produced pairs while in Unruh one over the excited states of the classical detector. The relative occupation number of two excited states is
Adiabaticity parameter • The adiabaticity parameter measures how rapidly the external field change • At early proper time τ<<1/Qs it follows from the parton model that γ<<1 since • Then the spectrum is: as it should be in E=const
Time evolution of pair production • At later time τ∼1/Qs the parton cascade decays and γ∼1 • The field E∝e-ωτ due to the screening of the original field by the field of produced pairs. • Momentum conservation (momentum lost by the fields equals momentum gain by produced particles) implies
Statistical interpretation • Consider the relative probability of single pair production • The probability that no pairs is produced w0 satisfies • Total probability that vacuum is unchanged is
Statistical interpretation • Narozhny and Nikishov demonstrated in 1970 that the imaginary part of the effective action for multi-pair production has the same form as the thermodynamic potential: • We can follow the produced system till times τ∼1/T. • However, we cannot follow it all the way to the equilibrium until we solve the back-reaction problem. see Cooper, Eisenberg, Kluger, Mottola, Svetitsky, hep-ph/9212206
Summary • We have started to reexamine the role of the vacuum instability in high energy QCD. • Due to existence of a hard scale Qs we hope to be able to analytically address many interesting problems. • Already now we can see that these new ideas can considerably modify our physical picture of particle production in high energy.